8655f7706f
This PR changes how Lean proves the equational theorems for structural recursion. The core idea is to let-bind the `f` argument to `brecOn` and rewriting `.brecOn` with an unfolding theorem. This means no extra case split for the `.rec` in `.brecOn` is needed, and `simp` doesn't change the `f` argument which can break the definitional equality with the defined function. With this, we can prove the unfolding theorem first, and derive the equational theorems from that, like for all other ways of defining recursive functions. Backs out the changes from #10415, the old strategy works well with the new goals. Fixes #5667 Fixes #10431 Fixes #10195 Fixes #2962
105 lines
2.3 KiB
Text
105 lines
2.3 KiB
Text
inductive N where
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| zero : N
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| succ : N → N
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def replace (f : N → Option N) (t : N) : N :=
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match f t with
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| some u => u
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| none =>
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match t with
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| .zero => .zero
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| .succ t' => replace f t'
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/--
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info: equations:
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@[defeq] theorem replace.eq_1 : ∀ (f : N → Option N),
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replace f N.zero =
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match f N.zero with
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| some u => u
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| none => N.zero
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@[defeq] theorem replace.eq_2 : ∀ (f : N → Option N) (t' : N),
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replace f t'.succ =
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match f t'.succ with
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| some u => u
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| none => replace f t'
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-/
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#guard_msgs in
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#print equations replace
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def replace2 (f : N → Option N) (t1 t2 : N) : N :=
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match f t1 with
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| some u => u
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| none =>
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match t2 with
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| .zero => .zero
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| .succ t' => replace2 f t1 t'
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/--
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info: equations:
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@[defeq] theorem replace2.eq_1 : ∀ (f : N → Option N) (t1 : N),
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replace2 f t1 N.zero =
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match f t1 with
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| some u => u
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| none => N.zero
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@[defeq] theorem replace2.eq_2 : ∀ (f : N → Option N) (t1 t' : N),
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replace2 f t1 t'.succ =
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match f t1 with
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| some u => u
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| none => replace2 f t1 t'
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-/
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#guard_msgs in
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#print equations replace2
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-- Now also mutual
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mutual
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inductive N1 where
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| zero : N1
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| succ : N2 → N1
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inductive N2 where
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| zero : N2
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| succ : N1 → N2
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end
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mutual
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def replaceMut1 (f : N1 → Option N1) (g : N2 → Option N2) (t : N1) : N1 :=
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match f t with
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| some u => u
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| none =>
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match t with
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| .zero => .zero
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| .succ t' => .succ (replaceMut2 f g t')
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def replaceMut2 (f : N1 → Option N1) (g : N2 → Option N2) (t : N2) : N2 :=
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match g t with
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| some u => u
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| none =>
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match t with
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| .zero => .zero
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| .succ t' => .succ (replaceMut1 f g t')
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end
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/--
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info: theorem replaceMut1.eq_def : ∀ (f : N1 → Option N1) (g : N2 → Option N2) (t : N1),
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replaceMut1 f g t =
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match f t with
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| some u => u
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| none =>
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match t with
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| N1.zero => N1.zero
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| N1.succ t' => N1.succ (replaceMut2 f g t')
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-/
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#guard_msgs in
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#print sig replaceMut1.eq_def
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/--
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info: theorem replaceMut2.eq_def : ∀ (f : N1 → Option N1) (g : N2 → Option N2) (t : N2),
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replaceMut2 f g t =
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match g t with
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| some u => u
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| none =>
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match t with
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| N2.zero => N2.zero
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| N2.succ t' => N2.succ (replaceMut1 f g t')
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-/
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#guard_msgs in
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#print sig replaceMut2.eq_def
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